Average-case ray shooting and minimum weight triangulations

Boris Aronov, Steven Fortune

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    Consider an environment filled with polyhedral obstacles. The answer to a ray-shooting query is the first obstacle encountered by the ray. A simple way to answer ray-shooting queries is to triangulate space in a manner compatible with the obstacles, and then walk through the triangulation along the ray until the first obstacle is encountered. We show that the average walk length can be reduced by choosing a triangulation with weight (i.e., length in two dimensions or area in three dimensions) as small as possible. In two dimensions, we observe that the length of the minimum-length triangulation can be estimated by the total length of the obstacles plus the total length of a minimum spanning tree of the obstacles, up to a logarithmic factor; this gives an a priori estimate of the average-case walk length. Our main result is in three dimensions. We give a polynomial-time algorithm that computes a triangulation compatible with a set of polyhedral obstacles; the area of the triangulation is within a multiplicative constant of the smallest possible.

    Original languageEnglish (US)
    Title of host publicationProceedings of the Annual Symposium on Computational Geometry
    Editors Anon
    PublisherACM
    Pages203-211
    Number of pages9
    StatePublished - 1997
    EventProceedings of the 1997 13th Annual Symposium on Computational Geometry - Nice, Fr
    Duration: Jun 4 1997Jun 6 1997

    Other

    OtherProceedings of the 1997 13th Annual Symposium on Computational Geometry
    CityNice, Fr
    Period6/4/976/6/97

    Fingerprint

    Ray Shooting
    Triangulation
    Walk
    Three-dimension
    Half line
    Two Dimensions
    Triangulate
    Query
    Minimum Spanning Tree
    Polynomials
    A Priori Estimates
    Polynomial-time Algorithm
    Multiplicative
    Logarithmic

    ASJC Scopus subject areas

    • Chemical Health and Safety
    • Software
    • Safety, Risk, Reliability and Quality
    • Geometry and Topology

    Cite this

    Aronov, B., & Fortune, S. (1997). Average-case ray shooting and minimum weight triangulations. In Anon (Ed.), Proceedings of the Annual Symposium on Computational Geometry (pp. 203-211). ACM.

    Average-case ray shooting and minimum weight triangulations. / Aronov, Boris; Fortune, Steven.

    Proceedings of the Annual Symposium on Computational Geometry. ed. / Anon. ACM, 1997. p. 203-211.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Aronov, B & Fortune, S 1997, Average-case ray shooting and minimum weight triangulations. in Anon (ed.), Proceedings of the Annual Symposium on Computational Geometry. ACM, pp. 203-211, Proceedings of the 1997 13th Annual Symposium on Computational Geometry, Nice, Fr, 6/4/97.
    Aronov B, Fortune S. Average-case ray shooting and minimum weight triangulations. In Anon, editor, Proceedings of the Annual Symposium on Computational Geometry. ACM. 1997. p. 203-211
    Aronov, Boris ; Fortune, Steven. / Average-case ray shooting and minimum weight triangulations. Proceedings of the Annual Symposium on Computational Geometry. editor / Anon. ACM, 1997. pp. 203-211
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